Determining a tangent space and filtering data onto a manifold

Abstract

A technique for determining the number of constraints on, or topological dimension of, a set of input data produced by a nonlinear system, such as a pathological vocal or econometric system. The technique characterizes the tangent space about a predetermined base point by identifying a maximal set of non-redundant nonlinear fits to the data. It needs only a few data points and only assumes that the functional form of the true constraints is smooth. Each fit is equivalent to a set of contours, with the data lying along the zero-value contour. For each fit, the gradient at the base point in the uphill direction identifies the constraint direction. The number of linearly independent constraint directions provides the number of constraints near the base point. The remaining unconstrained directions define the tangent space, which has a dimensionality equal to the number of linearly independent unconstrained directions.